# Homotopy Type Theory homotopy groups of spheres (Rev #9, changes)

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## Idea

The homotopy groups of spheres are a fundemental concept in algebraic topology. They tell you about homotopy classes of maps from spheres to other spheres which can be rephrased as the collection of different ways to attach a sphere to another sphere. The homotopy type of a CW complex is completely determined by the homotopy types of the attaching maps.

Here’s a quick reference for the state of the art on homotopy groups of spheres in HoTT. Everything listed here is also discussed on the page on Formalized Homotopy Theory.

## Table

$n\backslash k$01234
0$\pi_0(S^0)$$\pi_0(S^1)$$\pi_0(S^2)$$\pi_0(S^3)$$\pi_0(S^4)$
1$\pi_1(S^0)$$\pi_1(S^1)$$\pi_1(S^2)$$\pi_1(S^3)$$\pi_1(S^4)$
2$\pi_2(S^0)$$\pi_2(S^1)$$\pi_2(S^2)$$\pi_2(S^3)$$\pi_2(S^4)$
3$\pi_3(S^0)$$\pi_3(S^1)$$\pi_3(S^2)$$\pi_3(S^3)$$\pi_3(S^4)$
4$\pi_4(S^0)$$\pi_4(S^1)$$\pi_4(S^2)$$\pi_4(S^3)$$\pi_4(S^4)$

## In progress

### $\pi_4(S^3)$

• Guillaume has proved that there exists an $n$ such that $\pi_4(S^3)$ is $\mathbb{Z}_n$. Given a computational interpretation, we could run this proof and check that $n$ is 2. Added June 2016: Brunerie now has a proof that $n=2$, using cohomology calculations and a Gysin sequence argument.

## At least one proof has been formalized

### $\pi_3(S^2)$

• Peter L. has constructed the Hopf fibration as a dependent type. Lots of people around know the construction, but I don’t know anywhere it’s written up. Here’s some Agda code with it in it.
• Guillaume’s proof that the total space of the Hopf fibration is $S^3$, together with $\pi_n(S^n)$, imply this by a long-exact-sequence argument.
• This was formalized in Lean in 2016.

### $\pi_n(S^2)=\pi_n(S^3)$ for $n\geq 3$

• This follows from the Hopf fibration and long exact sequence of homotopy groups.
• It was formalized in Lean in 2016.

### Freudenthal Suspension Theorem

Implies $\pi_k(S^n) = \pi_{k+1}(S^{n+1})$ whenever $k \le 2n - 2$

• Peter’s encode/decode-style proof, formalized by Dan, using a clever lemma about maps out of two n-connected types.

### $\pi_2(S^2)$

• Guillaume’s proof using the total space the Hopf fibration.
• Dan’s encode/decode-style proof.

### $\pi_1(S^1)$

• Mike’s proof by contractibility of total space of universal cover (HoTT blog).
• Dan’s encode/decode-style proof (HoTT blog). A paper mostly about the encode/decode-style proof, but also describing the relationship between the two.
• Guillaume’s proof using the flattening lemma.

Revision on September 14, 2018 at 08:22:35 by Ali Caglayan. See the history of this page for a list of all contributions to it.